Earlier quoted context omitted.
Being Jewish is also multi-tiered because it's associated with both a religion and an ethnicity. I'd love to popularize being thought of Ashkenazi and not Jewish. I don't have a familial bond with Israel for probably 17 centuries and I don't really care if I ever did. My culture is more strongly associated with Eastern Europe and we were run out of town on a rail 100 years ago.
Ashkenazi is just an old hebrew word for German. If you strip away both Germany (the HRE, more specifically), and Judaism then the term doesn't have much meaning. At least, not as an identity that people have assumed historically. Ashkenazi Jews before the war just called themselves Jews, with secular emancipationists often appending nationality.
Naturally under the Ashkenazi umbrella there were people of different cultures (and different degrees of assimilation to the surrounding non-Jewish population), but those Jews were still more similar to one another than to non-Ashkenazi Jews.